Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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42 CHAPTER 3. BEHAVIOURAL EQUIVALENCES different deadlock behaviour when made to interact with other parallel processes— a highly undesirable state of affairs. In light of the above example, we are forced to reject the law α.(P + Q) = α.P + α.Q , which is familiar from the standard theory of regular languages, for our desired notion of behavioural equivalence. (Can you see why?) Therefore we need to refine our notion of equivalence in order to differentiate processes that, like the two vending machines above, exhibit different reactive behaviour while still having the same traces. Exercise 3.2 (Recommended) A completed trace of a process P is a sequence α1 · · · αk ∈ Act ∗ (k ≥ 0) such that there exists a sequence of transitions P = P0 α1 α2 αk → P1 → P2 · · · Pk−1 → Pk , for some P1, . . . , Pk. The completed traces of a process may be seen as capturing its deadlock behaviour, as they are precisely the sequences of actions that may lead the process into a state from which no further action is possible. 1. Do the processes and (CA | CTM) \ {coin, coffee, tea} (CA | CTM ′ ) \ {coin, coffee, tea} defined above have the same completed traces? 2. Is it true that if P and Q are two CCS processes affording the same completed traces and L is a set of labels, then P \ L and Q \ L also have the same completed traces? You should, of course, argue for your answers. 3.3 Strong bisimilarity Our aim in this section will be to present one of the key notions in the theory of processes, namely strong bisimulation. In order to motivate this notion intuitively, let us reconsider once more the two processes CTM and CTM ′ that we used above to argue that trace equivalence is not a suitable notion of behavioural equivalence for reactive systems. The problem was that, as fully formalized in Exercise 3.2, the
3.3. STRONG BISIMILARITY 43 trace equivalent processes CTM and CTM ′ exhibited different deadlock behaviour when made to interact with a third parallel process, namely CA. In hindsight, this is not overly surprising. In fact, when looking purely at the (completed) traces of a process, we focus only on the sequences of actions that the process may perform, but do not take into account the communication capabilities of the intermediate states that the process traverses as it computes. As the above example shows, the communication potential of the intermediate states does matter when we may interact with the process at all times. In particular, there is a crucial difference in the capabilities of the states reached by CTM and CTM ′ after these processes have received a coin as input. Indeed, after accepting a coin the machine CTM always enters a state in which it is willing to output both coffee and tea, depending on what its user wants, whereas the machine CTM ′ can only enter a state in which it is willing to deliver either coffee or tea, but not both. The lesson that we may learn from the above discussion is that a suitable notion of behavioural relation between reactive systems should allow us to distinguish processes that may have different deadlock potential when made to interact with other processes. Such a notion of behavioural relation must take into account the communication capabilities of the intermediate states that processes may reach as they compute. One way to ensure that this holds is to require that in order for two processes to be equivalent, not only should they afford the same traces, but, in some formal sense, the states that they reach should still be equivalent. You can easily convince yourselves that trace equivalence does not meet this latter requirement, as the states that CTM and CTM ′ may reach after receiving a coin as input are not trace equivalent. The classic notion of strong bisimulation equivalence, introduced by David Park in (Park, 1981) and widely popularized by Robin Milner in (Milner, 1989), formalizes the informal requirements introduced above in a very elegant way. Definition 3.2 [Strong bisimulation] A binary relation R over the set of states of an LTS is a bisimulation iff whenever s1 R s2 and α is an action: - if s1 α → s ′ 1 , then there is a transition s2 α → s ′ 2 such that s′ 1 R s′ 2 ; - if s2 α → s ′ 2 , then there is a transition s1 α → s ′ 1 such that s′ 1 R s′ 2 . Two states s and s ′ are bisimilar, written s ∼ s ′ , iff there is a bisimulation that relates them. Henceforth the relation ∼ will be referred to as strong bisimulation equivalence or strong bisimilarity. Since the operational semantics of CCS is given in terms of an LTS whose states are CCS process expressions, the above definition applies equally well to CCS
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Reactive Systems: Modelling, Specif
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iv CONTENTS 5 Hennessy-Milner logic
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List of Figures 2.1 The interface f
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Page 12 and 13: xii PREFACE It has been very satisf
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92 CHAPTER 4. THEORY OF FIXED POINT
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Chapter 5 Hennessy-Milner logic In
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103 We are interested in using the
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105 of the one step transitions a
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1. Decide whether the following sta
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109 Note that logical negation is n
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111 Another example of a process th
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a s s1 b s2 b a a t b t1 b t
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Chapter 6 Hennessy-Milner logic wit
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CHAPTER 6. HML WITH RECURSION 117 e
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CHAPTER 6. HML WITH RECURSION 119
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6.1. EXAMPLES OF RECURSIVE PROPERTI
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6.2. SYNTAX AND SEMANTICS OF HML WI
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6.2. SYNTAX AND SEMANTICS OF HML WI
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6.3. LARGEST FIXED POINTS AND INVAR
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6.4. GAME CHARACTERIZATION FOR HML
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6.5. MUTUALLY RECURSIVE EQUATIONAL
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6.5. MUTUALLY RECURSIVE EQUATIONAL
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6.6. CHARACTERISTIC PROPERTIES 139
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6.7. MIXING LARGEST AND LEAST FIXED
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6.8. FURTHER RESULTS ON MODEL CHECK
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Chapter 7 Modelling and analysis of
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CHAPTER 7. MODELLING MUTUAL EXCLUSI
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CHAPTER 7. MODELLING MUTUAL EXCLUSI
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7.1. SPECIFYING MUTUAL EXCLUSION IN
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7.2. SPECIFYING MUTUAL EXCLUSION US
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7.2. SPECIFYING MUTUAL EXCLUSION US
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7.3. TESTING MUTUAL EXCLUSION 169 i
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7.3. TESTING MUTUAL EXCLUSION 171 D
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